2.6 Modelling skillsIB Maths: Applications and Interpretation HL: Revision notes

Mathematical modelling turns a real situation into mathematics and back again. Follow the cycle:

1. Develop the model: understand the context, choose a function type and its parameters.
2. Fit the model: find the parameters from the data.
3. Test and reflect: compare the model with the data and the context.
4. Use the model: read values, interpret them and make predictions. If the model is not reasonable, return to step 1 and choose or adjust the model. You use the SL 2.5 models (linear, quadratic, exponential, direct and inverse variation, cubic, sinusoidal) and their graphs. Fitting a model by regression is topic 4, and at SL you are not expected to perform non-linear regressions.

Choose the model from the shape of the data, the properties of the curve and the context:

• constant rate of change: linear
• one maximum or minimum: quadratic
• the same multiplying factor for equal steps, or levelling off at an asymptote: exponential
• regular repeating cycle: sinusoidal. The domain is the set of inputs that make sense. Time cannot be negative, a tank cannot hold more than its capacity and a count cannot be fractional. Example: $V(t)=40+25t$ for a tank of capacity 400 litres gives $40+25t\leq400$, so $0\leq t\leq14.4$.

There are three ways to find the parameters. Initial conditions: for $f(t)=ka^t$, $f(0)=k$. For $a\sin(bt)+d$, $a=\frac{\max-\min}{2}$, $d=\frac{\max+\min}{2}$ and $b=\frac{360}{\text{period}}$. Substitution of points: put known points into the function. Through $(0,50)$ and $(3,400)$ with $f(t)=ka^t$: $k=50$ and $50a^3=400$, so $a=2$. Simultaneous equations (GDC): set up one equation for each unknown, up to three linear equations in three variables, and solve with technology. Example: $y=ax^2+bx+c$ through $(0,5)$, $(1,9)$ and $(3,29)$ gives $c=5$, $a+b+c=9$, $9a+3b+c=29$, so $a=2$, $b=2$, $c=5$.

Test the model against the data and the context. Comment on whether it is appropriate and reasonable:

• Does the curve pass close to the data points, and does it have the right shape?
• Do the predicted values make sense (no negative lengths, no population below zero, no volume above capacity)?
• Is the domain sensible? Justify a choice of model with specific features, for example 'the mass halves every 6 days, so an exponential model is appropriate'. If a model gives an unreasonable value, say which assumption fails.

Use the model to read values, solve equations on the GDC and interpret the answers in context with units. Interpolation is predicting within the range of the data and is usually reliable. Extrapolation is predicting outside the range of the data and can be very unreliable. Example: a quadratic $P(t)=-0.03t^2+1.5t+40$ fitted to a town's population from 2000 to 2020 predicts a population of zero at $t=69.3$. That is not realistic for a growing town, so the model should not be trusted far beyond 2020. A bacteria model $N=500(1.4)^t$ would also give impossible numbers for large $t$, because food and space run out.